Review for Test 2

1
Review for Test 2

• Scheduled for Wednesday, April 2
• Covers 1.8, 2.4, 3.1, 3.2, 3.3, and 3.4

2
Formal Framework for Induction

• You are asked to prove some statement X for all n
  ? S.
• The set S is some well-ordered subset of the
  integers of the form b, b 1, b 2, where
  b is an integer.
• The statement X is always some predicate
  involving n, such as 1 2 n n(n 1) /
  2.
• 1st step Introduce your predicate.
• ex Let P(n) denote 1 2 n n(n 1) /
  2.
• Now you desire to prove ?n ? S P(n). And the
  principle of mathematical induction tells you
  that you can do this by showing both P(b) and
  also ?n ? S P(n) ? P(n 1).

3

• 2nd step Prove base case Prove P(b)
• 3rd step Prove inductive case ?n ? S P(n) ?
  P(n 1)
• (a) Set up for universal generalization Let n
  ? S
• (b) Assume the hypothesis Assume P(n)
• (c) Show the consequent Show P(n 1)
• (d) Now by the principle of mathematical
  induction you have proved the desired result.

Know recursive definitions. If I give you a
recursive definition of a sequence or a set, you
should be able to write out the initial terms of
the sequence, or tell me what the set is equal
to. Ex -1 ? S. x y ? S whenever x ? S
and y ? S. Identify the set S. Answer S x ?
Z x lt 0 . Or S Z-. Be able to prove this if
asked to. Structural Induction.
4

• Know the definition of a map and a function
• Be able to determine if a map is a function
• Determine the domain, codomain, and range of a
  function
• Determine if a function is one-to-one and/or onto
• What is the significance of a one-to-one
  correspondence
• Inverse functions (construct them, when do they
  exist)
• Know how to compose two functions (when can you
  do it)
• What does it mean for a to divide b
• Be familiar with division theorems
• What is a prime number What is a composite
  number
• Fundamental Theorem of Arithmetic (know how to
  factor)
• There are infinitely many primes (could you prove
  this)
• The Division Algorithm
• gcd, lcm, how to exploit the prime factorizations
  to get them
• Modular arithmetic, congruences

5

• Proof strategies, working backwards, etc. The
  goal is that you can prove complex statements.
• When can you use a counter-example
• Sequences and Summations
• Know summation notation evaluate sums
• Know sequence notation given a sequence
  definition, produce terms of the sequence
• Work with summation notation
• What is the definition of a sequence
• How are sequences used in cardinality/countability
  proofs
• What does it mean for two sets to have the same
  cardinality
• What does it mean for a set to be countable or
  uncountable
• How to prove a set is countable
• How to prove a set is uncountable
• Note the following theorem that we never
  introduced, but some of you used it intuitively
• If f S ? T is onto, then S ? T

6
Diagonalization

• We used the technique of diagonalization to
  prove that the set of real numbers is
  uncountable.
• Use this technique to prove that P(N) is
  uncountable!
• Why cant we use diagonalization to show N is
  uncountable?
• Clearly we cant because N is countable
• We can enumerate N or construct a one-to-one
  correspondence between N and Z.
• Note that the definition of a countable set is
  that it is finite or can be put in one-to-one
  correspondence with Z.
• Actually all we need to do is put it in
  one-to-one correspondence with another set we
  know to be countable.